Stability of Functional Equations and Properties of Groups

Gian Luigi Forti

doi:10.2478/amsil-2019-0004

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Stability of Functional Equations and Properties of Groups

Annales Mathematicae Silesianae

Volume 33 (2019): Issue 1 (September 2019)

By: Gian Luigi Forti

Open Access

|Jul 2019

Abstract

Investigating Hyers–Ulam stability of the additive Cauchy equation with domain in a group G, in order to obtain an additive function approximating the given almost additive one we need some properties of G, starting from commutativity to others more sophisticated. The aim of this survey is to present these properties and compare, as far as possible, the classes of groups involved.

References

[1] R. Badora, On the Hahn–Banach theorem for groups, Arch. Math. (Basel) 86 (2006), no. 6, 517–528.10.1007/s00013-005-1570-0
Open DOI Search in Google Scholar Back to article
[2] R. Badora, B. Przebieracz, P. Volkmann, On Tabor groupoids and stability of some functional equations, Aequationes Math. 87 (2014), no. 1–2, 165–171.10.1007/s00010-013-0206-x
Open DOI Search in Google Scholar Back to article
[3] A. Bahyrycz, Forti’s example of an unstable homomorphism equation, Aequationes Math. 74 (2007), no. 3, 310–313.10.1007/s00010-007-2882-x
Open DOI Search in Google Scholar Back to article
[4] J. Dixmier, Les moyennes invariantes dans les semi-groups et leurs applications, Acta. Sci. Math. Szeged 12 (1950), 213–227.
Search in Google Scholar Back to article
[5] G.-L. Forti, The stability of homomorphisms and amenability, with applications to functional equations, Abh. Math. Sem. Univ. Hamburg 57 (1987), 215–226.10.1007/BF02941612
Open DOI Search in Google Scholar Back to article
[6] G.-L. Forti, Hyers–Ulam stability of functional equations in several variables, Aequationes Math. 50 (1995), no. 1–2, 143–190.10.1007/BF01831117
Open DOI Search in Google Scholar Back to article
[7] G.-L. Forti, J. Sikorska, Variations on the Drygas equation and its stability, Nonlinear Anal. 74 (2011), no. 2, 343–350.10.1016/j.na.2010.08.004
Open DOI Search in Google Scholar Back to article
[8] Z. Gajda, Z. Kominek, On separation theorems for subadditive and superadditive functionals, Studia Math. 100 (1991), no. 1, 25–38.10.4064/sm-100-1-25-38
Open DOI Search in Google Scholar Back to article
[9] F.P. Greenleaf, Invariant Means on Topological Groups, Van Nostrand Mathematical Studies 16, Van Nostrand Reinhold Co., New York–Toronto–London–Melbourne, 1969.
Search in Google Scholar Back to article
[10] D.H. Hyers, On the stability of the linear functional equation, Proc. Nat. Acad. Sci. U.S.A. 27 (1941), 222–224.10.1073/pnas.27.4.222107831016578012
Open DOI Search in Google Scholar Back to article
[11] S.V. Ivanov, The free Burnside groups of sufficiently large exponents, Internat. J. Algebra Comput. 4 (1994), no. 1–2, 1–308.10.1142/S0218196794000026
Open DOI Search in Google Scholar Back to article
[12] W. Magnus, A. Karrass, D. Solitar, Combinatorial Group Theory, Dover Publications, Inc., New York, 1976.
Search in Google Scholar Back to article
[13] P.S. Novikov, S.I. Adian, Infinite periodic groups. I–III. (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 32 (1968), 212–244, 251–254, 709–731.
Search in Google Scholar Back to article
[14] A.Yu. Ol’shanskii, On the question of the existence of an invariant mean on a group. (Russian), Uspekhi Mat. Nauk 35 (1980), no. 4(214), 199–200.10.1070/RM1980v035n04ABEH001876
Open DOI Search in Google Scholar Back to article
[15] A.Yu. Ol’shanskii, The Novikov–Adian theorem. (Russian), Mat. Sb. (N.S.) 118(160) (1982), no. 2, 203–235.10.1070/SM1983v046n02ABEH002768
Open DOI Search in Google Scholar Back to article
[16] D.V. Osin, Uniform non-amenability of free Burnside groups, Arch. Math. (Basel) 88 (2007), no. 5, 403–412.10.1007/s00013-006-2002-5
Open DOI Search in Google Scholar Back to article
[17] J. Rätz, On approximately additive mappings, in: E.F. Beckenbach (ed.), General Inequalities. 2, Proc. Second Internat. Conf., Oberwolfach 1978, Birkhäuser, Basel, 1980, pp. 233–251.10.1007/978-3-0348-6324-7_22
Open DOI Search in Google Scholar Back to article
[18] E. Shulman, Group representations and stability of functional equations, J. London Math. Soc. (2) 54 (1996), no. 1, 111–120.10.1112/jlms/54.1.111
Open DOI Search in Google Scholar Back to article
[19] E. Shulman, Addition theorems and related geometric problems of group representation theory, in: J. Brzd¦k et al. (eds.), Recent Developments in Functional Equations and Inequalities, Banach Center Publ., vol. 99, Polish Acad. Sci. Inst. Math., Warsaw, 2013, pp. 155–172.10.4064/bc99-0-10
Open DOI Search in Google Scholar Back to article
[20] F. Skof, Proprietà locali e approssimazione di operatori. Geometry of Banach spaces and related topics (Milan, 1983), Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129 (1986).10.1007/BF02924890
Open DOI Search in Google Scholar Back to article
[21] L. Székelyhidi, Note on a stability theorem, Canad. Math. Bull. 25 (1982), no. 4, 500–501.10.4153/CMB-1982-074-0
Open DOI Search in Google Scholar Back to article
[22] L. Székelyhidi, Remark 17, 22sd International Symposium on Functional Equations, Oberwolfach 1984, Aequationes Math. 29 (1985), no. 1, 95.
Search in Google Scholar Back to article
[23] J. Tabor, Remark 18, 22sd International Symposium on Functional Equations, Oberwolfach 1984, Aequationes Math. 29 (1985), no. 1, 96.
Search in Google Scholar Back to article
[24] I. Toborg, Tabor groups with finiteness conditions, Aequationes Math. 90 (2016), no. 4, 699–704.10.1007/s00010-015-0369-8
Open DOI Search in Google Scholar Back to article
[25] D. Yang, Remarks on the stability of Drygas’ equation and the Pexider–quadratic equation, Aequationes Math. 68 (2004), no. 1–2, 108–116.10.1007/s00209-003-0573-4
Open DOI Search in Google Scholar Back to article
[26] D. Yang, The stability of the quadratic functional equation on amenable groups, J. Math. Anal. Appl. 291 (2004), no. 2, 666–672.10.1016/j.jmaa.2003.11.021
Open DOI Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/amsil-2019-0004 | Journal eISSN: 2391-4238 (formerly 0860-2107) | Journal ISSN: 0860-2107

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Language: English

Page range: 77 - 96

Submitted on: Mar 18, 2019

Accepted on: Apr 15, 2019

Published on: Jul 18, 2019

Published by: University of Silesia in Katowice, Institute of Mathematics

In partnership with: Paradigm Publishing Services

Keywords:

additive functional equation,

quadratic functional equation,

stability,

amenable group,

weak commutativity

MSC:

39B82,

20F12,

43A07

Related subjects:

Mathematics,

General mathematics

© 2019 Gian Luigi Forti, published by University of Silesia in Katowice, Institute of Mathematics
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Volume 33 (2019): Issue 1 (September 2019)